The NLSE is the simplest framework to describe the wave propagation of nonlinear dispersive waves [1]}. For the deep-water wave problem the slowly-varying wave envelope \(\Psi \)  around the wave number \(k\)  satisfies [2]}, [3]}
\(i\left(\Psi _x+\dfrac{2k}{\omega }\Psi _t\right)-\dfrac{k^2}{\omega }\Psi _{tt}-k^3\left|\Psi \right|^2\Psi =0,\) 
